1

TAYLOR'S POWER LAW: BEFORE AND AFTER 50 YEARS OF

SCIENTIFIC SCRUTITY

XU MENG1,*

1 Department of Mathematics and Physics, University of New Haven, 300 Boston Post Road,

West Haven, CT 06516. United States of America

*Corresponding author

Email: mxu@newhaven.edu

Abstract. Taylor's power law is one of the mostly widely known empirical patterns in ecology discovered

in the 20th century. It states that the variance of species population density scales as a power-law function

of the mean population density. Taylor's power law was named after the British ecologist Lionel Roy

Taylor. During the past half-century, Taylor's power law was confirmed for thousands of biological

species and even for non-biological quantities. Numerous theories and models have been proposed to

explain the mechanisms of Taylor's power law. However an understanding of the historical origin of this

ubiquitous scaling pattern is lacking. This work reviews two research aspects that are fundamental to the

discovery of Taylor's power law and provides an outlook of its future studies.

Keywords: L. R. Taylor; logarithm; mean; scaling; variance

2

Introduction

Lionel Roy Taylor (1914-2007), a British ecologist, published an article entitled

"Aggregation, variance and the mean" in the March 4, 1961 issue of Nature. In the

paper, Taylor studied the relationship between the mean and the variance of population

density for at least 22 biological species in 24 data sets (Taylor 1961, referred to as

TL61 hereafter). On the log-log scale, Taylor showed that the mean-variance

relationship was described well by a linear equation

log(𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒) = log 𝑎 + 𝑏 log(𝑚𝑒𝑎𝑛) , 𝑎 > 0. (eqn 1)

On arithmetic scale, eqn 1 becomes a power law

𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝑎(𝑚𝑒𝑎𝑛)𝑏 , 𝑎 > 0. (eqn 2)

Eqn 2 earned the name "Taylor's law" (not to be confused with "Taylor law", the

Public Employees Fair Employment Act) or "Taylor's power law (TPL)" after extensive

empirical confirmations using aphid, moth, and bird data collected in England by Taylor

and his colleagues (Taylor, Woiwod, Perry 1978; Taylor, Woiwod 1980; Taylor,

Woiwod 1982; Taylor, Taylor, Woiwod, Perry 1983; Taylor 1984). The exact phrase

"Taylor's power law" first appeared in the paper by D. G. Harcourt (1963), where the

author studied the spatial distribution of Colorado beetle, a pest in potato crops. In the

recent decades, studies on TPL were reinvigorated by the increased power of statistical

computation and improved accessibility to ecological data sets (Brown et al. 2002;

Marquet et al. 2005; Eisler, Bartos, Kertész 2008). TL61, widely perceived as the

founding work of TPL, has been cited in roughly 2174 literature up to today (last

retrieved from Google Scholar on 04/07/2016).

In the past half-century, both theoretical and applied studies were conducted on TPL.

Numerous biological and statistical models have been proposed to explain the

underlying mechanisms of TPL (Taylor, Taylor 1977; Hanski 1987; Perry 1988;

Kilpatrick, Ives 2003; Cohen, Xu, Schuster 2013; Giometto et al. 2015; Cohen, Xu 2015;

Xiao et al. 2015). A list of physical models relevant to TPL can be found in Eisler et al.

(2008). In practice, TPL has been applied to the sampling design of agricultural pest

(Green 1970; Kuno 1991; Celini, Vaillant 2004), sample size estimation of fish species

(Mouillot et al. 1999), and quantification of temporal variability in seabird (Certain et al.

2007) and reef fish (Mellin et al. 2010) populations.

TPL was debated on its mathematical formulation and ecological implications.

Several competing models of TPL have been proposed based on their statistical

superiority in data fitting (Routledge, Swartz 1991; Perry, Woiwod 1992; Tokeshi 1995).

Other scientists questioned TPL's biological interpretation and usefulness (Anderson et

al. 1982; Downing 1986; Kuno 1991). These critiques propelled the theoretical

development of TPL, however a convincing and unified theory of TPL is still lacking.

Until such a theory can be found or the contradicting evidences of TPL (Anderson et al.

1982; Downing 1986) can be validated, understanding of this ubiquitous pattern remains

incomplete.

3

This paper reviews the historical developments of mean-variance scaling before and

after TL61, and highlights several unsolved research questions of TPL. To the author's

knowledge, the history of TPL has not been thoroughly investigated in the existing

literature. For modern ecologists, testing and modeling empirical patterns have become

a major component of their work, while it is equally important to appreciate and beware

of the scientific roots of established concepts and models. Through this retrospective

thinking the limitation of examined patterns can be exposed, a prudent model-building

approach can be taken, and unsolved open questions can be revealed. It is the hope of

this work that it will raise the awareness of the historical developments of TPL among

quantitative ecologists, and aid their research on this ubiquitous empirical pattern.

I review the research of mean-variance scaling before TL61 and TPL after TL61 in

sections 2 and 3 respectively. In section 4 I present a statistical model that is

conceptually equivalent to but does not explain TPL. In section 5 I discuss some key

issues that have been overlooked in the research of TPL.

Mean-variance scaling before TL61

Studies on the mean-variance scaling began even before the term "variance" was

invented (Fisher 1918). In his 1879 article on thermal aerodynamics (Reynolds 1879),

the famous physicist Osborne Reynolds plotted the pressure difference against pressure

on the log-log axes, and compared the difference between hydrogen and air. Had

Reynolds used variance to quantify pressure differences, the origin of mean-variance

scaling would be traced back before any statistician or biologist started pursuing this

topic. The theory underlying Reynolds' plot (Fig. 1a) is beyond the expertise of the

author and therefore not described here.

In statistics, the correlation between the sample mean and higher-order sample

moments was derived in an editorial work of Biometrika (author unknown 1903, pp.

279), although the same result had been observed by Pearson and Filon (1897, pp. 236).

Neyman (1926, pp. 402) derived the regression coefficient between the sample variance

(dependent variable) and the sample mean (independent variable).

Mean-variance relationship gained broad scientific attentions when analysis of

variance (ANOVA, Fisher 1918, 1921) was developed and applied in natural sciences.

One of the key assumptions in ANOVA was the equality of variances across different

experimental groups (homoscedasticity). However, for distributions of most real-world

data, variance is not a constant but varies with the mean. To apply ANOVA, scientists

studied mean-variance scaling and designed variable transformations that can stabilize

the variance. For example, Bartlett (1936a) applied squared root transformation on

Poisson random variable to derive a constant variance. In a separate work, Bartlett

(1936b) proposed an analytic relationship between the sample mean and the sample

variance:

𝑠𝑎𝑚𝑝𝑙𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝛼(𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛) + 𝛽(𝑠𝑎𝑚𝑝𝑙𝑒 𝑚𝑒𝑎𝑛)2. (eqn 3)

Bartlett used eqn 3 to quantify variance of crane fly (Nephrotoma appendiculata)

larvae populations, and showed that it provided a better fit than Poisson, which leads to

mean = variance. Fisher et al. (1921) also used eqn 3 to describe bacterial population

4

densities. Clapham (1936) found that distributions of individual plants in prairie were

not Poisson but "over-dispersed" (variance > mean). Five years later, Bliss (1941) used

the log-linear form of TPL (eqn 1) to fit spatially grouped populations of Japanese

beetle larvae. Bliss's article was the first to publish the current form of TPL and pushed

the origin of TPL 20 years before Taylor's work in 19611. Later, Beall (1942)

systematically studied the mean-variance relationship. He noticed that in entomological

field data, "the departure of s2 (sample variance) from �̅� (sample mean) becomes

disproportionately great as �̅� increases", and used the functional mean-variance

relationship of a negative binomial distribution to account for this discrepancy. Namely,

𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 𝑚𝑒𝑎𝑛 + 𝑘(𝑚𝑒𝑎𝑛)2. (eqn 4)

Fracker and Brischle (1944) pointed out that eqn 4 was inadequate to describe ribes

populations because k changed with the mean and the quadrat sizes. They found

Bartlett's equation (eqn 3) and TPL (eqn 2) yielded better fits.

Entering 1950's, researchers continued to analyze the mean-variance scaling and to

explore different variance stabilizing transformations of interested variables, such as

tasters' scores on food (Hopkins 1950) and drink (Coote 1956), and catches of marine

species (Barnes 1952). Taylor (1961), using the power-law relationship (eqn 2) tested

for 24 ecological population data sets, defined a transformation that stabilizes the

variance as a constant (see eqn (3) in TL61). Unfortunately, the transformation formula

in TL61 was incomprehensible because Taylor did not provide any mathematical

derivations. Below I included a short proof of Taylor's formula.

For a random variable X, a variance-stabilizing function f is defined such that, for a

constant Q,

𝑠2(𝑓(𝑋)) = 𝑄.

Using the first-order delta method (Oehlet 1991),

𝑄 = 𝑠2(𝑓(𝑋)) ≈ 𝑠2(𝑋) ⋅ [𝑓′(𝑚)]2.

If TPL held with parameters a and b as specified in eqn 2, then above equation

becomes

𝑎𝑚𝑏 ⋅ [𝑓′(𝑚)]2 = 𝑄,

or

1 Eisler et al. (2008) reported that TPL was first discovered by H. Fairfield Smith

(1938). However a re-examination of Smith's paper showed that the author did not study

the relationship between variance of yield of agricultural crops and the corresponding

mean yield, but the relationship between the variance of yield and plot size.

5

𝑓′(𝑚) = (𝑄

𝑎)

12𝑚−

𝑏2.

Integrating both sides of the above equation yields

𝑓(𝑚) = (𝑄

𝑎)

1

2∫ 𝑢−

𝑏

2𝑚

𝑡𝑑𝑢. (eqn 5)

Eqn 5 is identical to eqn (3) in TL61. The form of f depends on the power exponent b

of TPL and t. For example, when b = 1 and t ≥ 0, f is a square root function. When b = 2

and t > 0, f is a natural logarithmic function. When b = 4, t ≠ 0, f(x) is a linear function

of 1/x.

Taylor's power law after TL61

The significance of TL61 does not necessarily lie in its discovery of the power-law

mean and variance scaling of species population densities, but in that it is the first meta-

analysis confirming TPL and establishing it as one of the quantitative patterns in

ecology (Smith et al. 2014). In the past half-century, Taylor's pioneer work has inspired

many biologists to test TPL against thousands of biological taxa. Such examples can be

found in the review by Eisler et al. (2008). As more and more empirical support of TPL

was found, the scientific interest of TPL has shifted from its empirical confirmation to

its underlying mechanisms. In this section, I will briefly discuss the theoretical

development of TPL in the past few decades, especially on the biological and statistical

models of TPL.

Three important works addressed the biological mechanisms of TPL from different

perspectives. Taylor and Taylor (1977) used density-dependent power law functions

that model animal migratory behaviors and account for TPL. Anderson et al. (1982), on

the other hand, illustrated TPL using classic population models with demographic and

environmental stochasticity, without incorporating any behavioral mechanism. Based on

a stochastic logistic model of multiple species, Kilpatrick and Ives (2003) showed that

increased interspecific competition reduced the slope of TPL to less than two. Several

other authors (Perry 1988, 1994; Ballantyne 2005) also have used population dynamic

models to explain TPL and its parameter values.

Recently, the statistical reasons of TPL have gained considerable attention, because

the large variety of taxa and models confirming TPL indicated that a unified

mechanism-independent theory must at work (Cohen and Xu 2015). Among several

explanations are the skewed distribution theory (Cohen and Xu 2015), plausible set

theory (Xiao et al. 2015), and large deviation theory (Giometto et al. 2015). The

common theme in these works is that the appearance of TPL does not rely on specific

biological processes. This school of thoughts provides new perspectives in tackling

TPL’s ubiquity in nature.

Tweedie's distributions and TPL

15 years before TL61, the British statistician, Maurice C.K. Tweedie (1946), asked

the question: "How can one determine the types of distribution in which the regression

6

function (sample variance as a function of sample mean) has a specified polynomial

form?" Tweedie was the first to explore the relationship between underlying probability

distribution and mean-variance scaling (when the polynomial function becomes a power

law). An important contribution by Tweedie was that he defined an "exponential family

(or Laplacian distribution)" for which the variance can be written as a polynomial

function of the mean (Tweedie 1947). A similar study when the variance was a

quadratic function of the mean was conducted by Morris (1982). Tweedie (1984) gave

special attention to a class of Laplacian distribution F(α) such that

𝑑(ln 𝑘2)

𝑑(ln 𝑘1)

is a constant independent of the distribution parameter α. Here k1 and k2 are the first and

second cumulants of the distribution respectively. Using the definition of cumulants, the

above quantity becomes

𝐹′′(𝛼) = 𝑎(−𝐹′(𝛼))𝑏

Interestingly, this equation is equivalent to the form of TPL (eqn 2) and has the

solution

𝐹(𝛼)

=

{

1

𝑎(−1)𝑏(2 − 𝑏)[(𝑎(−1)𝑏(1 − 𝑏)𝛼 + 𝑐1

1−𝑏)2−𝑏1−𝑏 − 𝑐1

2−𝑏] + 𝑐2 𝑓𝑜𝑟 𝑏 ≠ 1 𝑜𝑟 2

𝑐1𝑎(1 − 𝑒−𝑎𝛼) + 𝑐2 𝑓𝑜𝑟 𝑏 = 1

−log(𝑎 − 𝑐1𝑎𝛼)

𝑎+ 𝑐2 𝑓𝑜𝑟 𝑏 = 2

where c1 = F'(0) and c2 = F(0) (Tweedie 1984, pp. 582). Jørgensen (1997) gave a

mathematically equivalent solution. To summarize, Tweedie found explicitly a family

of distribution functions that satisfied TPL.

Kendal and colleagues (2004, 2011) argued that Tweedie's model provides a

universal explanation of TPL and defined a family of distributions that satisfy TPL, "the

Tweedie distributions". Several fundamental caveats exist in their theory. First, Kendal

et al. failed to notice that Tweedie's approach works for a polynomial relationship

between variance and mean, not only for the mean-variance power law described by

TPL. Second, in all claims made by Kendal and his colleagues, TPL (or "invariance

under scale transformation") was assumed a priori as a given assumption (Kendal 2004,

pp. 202; Kendal, Jørgensen 2011, pp066115-2, 3). This misuse of TPL by assuming its

validity, instead of deriving it, disproves the Tweedie's model as an explanation of TPL.

Third, as pointed out by Cohen et al. (2013), Tweedie distributions do not explain TPL

with power exponent between 0 and 1, which were found in some empirical studies

(Green 1970; Keil et al. 2010, Fig. 2). Overall, "Tweedie distributions" or "Tweedie's

model" do not provide a universal explanation of TPL, but merely incorporated a class

of well-known probability distributions for which the variance can be written as a

power-law of the mean, with a restricted range in the power-law exponent (see Table 1

in Kendal 2004, pp. 203).

7

Conclusions and outlooks of TPL research

My review showed that the development and application of ANOVA set up the

theoretical framework and necessity for the discovery of TPL. TPL was neither

universal nor superior to other mean-variance scaling relationships (Downing 1986;

Routledge and Swartz 1991) in the history. The meaning and usefulness of TPL rely on

confirmation against empirical data, which in turn requires new ways of data synthesis

and analysis.

As stated in the Introduction, one of the challenges in the research of TPL is that no

unified theory or model has yet been found. Most existing models of TPL relied on the

biological features of particular species or specific environmental and experimental

conditions, and lacked generality to be useful for various species across multiple scales.

Such problem results from the limited scope and scale of population abundance data

used in the testing of TPL. Taylor and colleagues confirmed TPL using aphid, insect

and bird population data collected from the Rothamsted Experimental Station (currently

Rothamsted Research) and throughout Great Britain (see Fig. 2 in Taylor, Taylor 1977

and Fig. 1 in Taylor, Woiwod 1980). However the species and geographical range

examined by Taylor may be limited to prove the universality of TPL. A synthetic

analysis of TPL using multiple large-scale data sets is therefore necessary to reveal

important properties of TPL (e.g. scale dependence, species differences) that may be

unobservable at a local scale. The following sections describe three current issues in the

studies of TPL and elaborate how multi-scale data analysis will help resolve these issues.

Scales in TPL

If an empirical pattern depended on specific temporal or spatial scales on which it

was tested, then the generality of this pattern and its usefulness should be questioned.

The testing of TPL relied on three scale measures: First, the size of quadrat or sampling

site, which is the area of habitat that the species lives and determines how many

individuals are included to compute the population count. Second, the number of

quadrats or sites within a block (or a super-cluster of quadrats or sites), or sample size,

which affects the accuracy in the estimations of the mean and the variance. Third, the

number of blocks used in the statistical fit of TPL, which reveals the overall scale of the

study site. All three scale measures contain information on the size and location of

spatial units, and their changes will lead to statistical or biological consequences to the

estimation of TPL parameters.

To examine the relation between scales and TPL, a naïve combination of data from

multiple experiments at various scales will not work, since the observed scale effects are

likely to be confounded with specific experimental methods or environmental

conditions. Existing literature studied the effect of quadrat size (Sawyer 1989) and

sample size (Clark, Perry 1994) on TPL and its parameters (slope and intercept in eqn 3).

However in both works the conclusions were based on simulated population counts

instead of real empirical data, and therefore cannot be applied to realistic ecological

scenarios. In fact, the main reason Sawyer (1989) used a simulation approach, as he

claimed, was the limitations of multiple scales in the ecological data sets. Such caveat

can be overcome with the use of multi-scale data that are currently available (e.g.

Breeding Bird Survey, Forest Inventory Analysis, North America Butterfly Association).

8

Estimation of mean and variance

The sample mean and sample variance of a species population exhibit interesting

self-restraint properties that may affect the parameters of TPL. For examples, in a

sample of population counts of size n, the sample variance is limited between 0 and nx̄2

(here x̄ being the sample mean, see Tokeshi 1995). Another issue in the testing of TPL

is the underestimation of population variance (and of population mean, to a less extent)

of skewed distributions (Ross 1990), which are often observed in species population

counts. Underestimated mean or variance will distort the true behavior of TPL using

population mean and variance, and yield uninterpretable statistical artifacts. While the

actual impact of this phenomenon on TPL and its parameters remains to be seen,

ecological data sets of large sample sizes may mitigate this issue to some extent.

Species specificity

Are the parameters of TPL specific to species? The most important work that addressed

this question was written by Downing (1986), where the author showed that the values

of exponent b of TPL may be similar among different species but vary according to

environments. Using published data, Downing also showed that the size of scale

measures (see Scales in TPL) affected the values of b, casting doubt to the applicability

of TPL. Taylor and colleagues (1988) disputed Downing's finding on its statistical

method and data quality, but did not provide a strategy to examine the species

specificity of the parameters of TPL. Meta-analysis of population data with comparable

scale measures could potentially answer this fundamental question.

Literature Search

Literature on mean-variance scaling and the use of logarithm in bivariate studies

were searched using exact phrases "variance function", "variance law", "mean-variance",

"mean and variance", "analysis of variance", "allometry", "logarithmic transformation",

and "logarithmic scale" as the topics of journal articles published before 1961 in the

Web of Science database and again in Google Scholar. Same phrases were also searched

in the archives of three oldest statistical journals in the world: Journal of the Royal

Statistical Society (first issue in 1838), Journal of the American Statistical Association

(first issue in 1888), and Biometrika (first issue in 1901) for relevant literature. Cross-

referencing was conducted in relevant articles.

Acknowledgements

The author thanks Joel E. Cohen for constructive comments and suggestions on an

earlier draft of the manuscript. The research was supported by University of New Haven

Summer Research Fund and NSF Grant No. 1038337 to the Rockefeller University.

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Figure 1. The woodcut figure from Reynold (1879) plotting logarithmic

difference in pressure against logarithmic pressure.